Random projections reduce the dimension of a set of vectors while preserving structural information, such as distances between vectors in the set. This paper proposes a novel use of row-product random matrices in random projection, where we call it Tensor Random Projection (TRP). It requires substantially less memory than existing dimension reduction maps. The TRP map is formed as the Khatri-Rao product of several smaller random projections, and is compatible with any base random projection including sparse maps, which enable dimension reduction with very low query cost and no floating point operations. We also develop a reduced variance extension. We provide a theoretical analysis of the bias and variance of the TRP, and a non-asymptotic error analysis for a TRP composed of two smaller maps. Experiments on both synthetic and MNIST data show that our method performs as well as conventional methods with substantially less storage.
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